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Prey-predator model with nonlocal and global consumption in the prey dynamics

  • * Corresponding author: Vitaly Volpert

    * Corresponding author: Vitaly Volpert

The third author is supported by "RUDN University Program 5-100"

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  • A prey-predator model with a nonlocal or global consumption of resources by prey is studied. Linear stability analysis about the homogeneous in space stationary solution is carried out to determine the conditions of the bifurcation of stationary and moving pulses in the case of global consumption. Their existence is confirmed in numerical simulations. Periodic travelling waves and multiple pulses are observed for the nonlocal consumption.

    Mathematics Subject Classification: Primary: 35K57.

    Citation:

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  • Figure 1.  Single pulse solution for prey and predator population for $ b = 10, d_1 = 0.1, d_2 = 0.1 $

    Figure 2.  Moving pulse for $ b = 10, d_1 = 0.3, d_2 = 0.1 $ (a) after time $ t = 500 $; (b) after time $ t = 600 $; (c) $ x $-$ t $ profile for $ L = 20 $

    Figure 3.  For two bifurcation diagrams, $ a = 1 $, $ \sigma_1 = 0.1 $, $ k_1 = k_2 = 0.35 $, $ \sigma_2 = 0.2 $ are fixed. (a) Bifurcation diagram in $ (L,d_1) $-plane for the values of parameters $ b = 5 $, $ \alpha = \beta = 0.35 $, $ d_2 = 0.1 $. (b) Bifurcation diagram in $ (d_1,b) $-plane for the values of other parameters $ \alpha = \beta = 0.363 $, $ d_2 = 1 $

    Figure 4.  Periodic travelling wave in the case of nonlocal consumption followed by a spatio-temporal structure. Left: a snapshot of solution with prey (blue) and predator (red) distributions. Right: level lines of the prey distribution $ u(x,t) $. The values of other parameters are $ a = 1,b = 1, \alpha = \beta = 0.363, \sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_1 = d_2 = 1 $

    Figure 5.  Multiple moving pulses in the case of nonlocal consumption. Left: a snapshot of solution with prey (blue) and predator (red) distributions. Right: level lines of the prey distribution $ u(x,t) $. The values of other parameters are $ a = 1,b = 1, \alpha = \beta = 0.375, \sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_1 = d_2 = 1 $

    Figure 6.  Different regimes observed in the case of nonlocal consumption presented on the $ (d_1,N) $ parameter plane for $ \alpha = 0.35 $ (left) and $ (N,\alpha) $ parameter plane for $ d_1 = 1 $ (right). The values of other parameters are $ a = 1,b = 1,\sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_2 = 1 $

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